The diophantine-equations tag has no wiki summary.

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### Prove a parametrization function is surjective

As a starting note, I would like to say that I haven't (yet) taken courses in Set Theory, so some higher-level notation may be lost on me (and I may not write everything conventionally), but I'll do ...

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### Does the following Diophantine equation have nontrivial rational solutions?

Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?
If I am ...

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### Lower bounds on the easier Waring problem

The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm ...

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### Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...

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### The diophantine eq. $x^4 +y^4 +1=z^2$

This question is an exact duplicate of the question
Does the equation $x^4+y^4+1=z^2$ have a non-trivial solution?
posted by Tito Piezas III on math.stackexchange.com.
The background of ...

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### Slight Alteration to a Diophantine Result

Hello all!
In the pursuit of a minor research problem I was pointed in the direction of an interesting result in the realm of Diophantine Analysis. The content of the result follows:
...

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### Counting and summing over solutions of a Diophantine equation

Say I have a Diophantine equation of the form $a_1 x_1 + a_2 x_2 + ... + a_m x_m = n$ such that the $a_is$ are all co-prime to each other. And I also have a function say $f$ which depends only on the ...

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### A remark of Mordell alluding to a local/global principle for cubic Diophantine equations

In Mordell Diophantine Equations he says:
In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of ...

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### Integer solutions of $n^k+(n+1)^k+\cdots+(n+m)^k=(n+m+1)^k$

So we all know already that next identities follow:
$3^2+4^2=5^2$
$3^3+4^3+5^3=6^3$
So it raises my question:
For $(*)n^k+(n+1)^k+...+(n+m)^k=(n+m+1)^k$ are there infinite triples (n,m,k) s.t ...

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### Advances and difficulties in effective version of Thue-Roth-Siegel Theorem

A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real ...

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### Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...

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### Is (n,m)=(18,7) the only positive solution to n^2 + n + 1 = m^3 ?

It's hard to do a Google search on this problem.
If I was using Maple correctly, there are no other positive solutions with n at most 10000.
I know some of these Diophantine questions succumb to ...

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### Fermat's Last Theorem in the cyclotomic integers.

Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am ...

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### Diophantine problem

I have reduced a knotty research problem to the following reasonable looking form:
Given any two integers $a$ and $b$, show that there are natural numbers $x_1,x_2,x_3$ and a (probaby negative) ...

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### Number Theory Representation of Primes

For a primes $p$ sufficiently large, does there always exists
positive integers $k,a,b\in\mathbb{N}$ such that $p=(k+1)(ab)+k(a+b)$ or equivalently
$p\equiv (ab)\bmod ((a+b)+ab)$?
Please note that ...

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### Which types of Diophantine equations are solvable?

Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we ...

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### Infinitely many solutions of a diophantine equation

If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...

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### The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $

Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem.
This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...

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### Methods for solving Pell's equation?

It is known that the minimum solution of Pell's equation $x^2-dy^2=\pm1$ can be found from the continued fraction expansion of $\sqrt d$. Are there other methods for finding the minimum (or any other) ...

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### Decision Procedure for Inequalities between Homogeneous Polynomials

Given two polynomials $p_1$ and $p2$ each of which is a multi-variate polynomial with positive integer coefficients, we want to decide if $p_1 \leq p_2$ over all integral values of the variables.
The ...

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### Diophantine equation with no integer solutions, but with solutions modulo every integer

It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...

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### English translation of Voronoi's dissertation

I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.

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### Analysis of a quadratic diophantine equation

Hi! This is my first post on Math Overflow. I have two equations: $a(3a-1) + b(3b-1) = c(3c-1)$ and $a(3a-1) - b(3b-1) = d(3d-1)$. I'm trying to find properties of $a$ and $b$ that lead to solutions, ...

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### Why certain diophantine equations are interesting (and others are not) ?

It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...

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### Is there an elementary way to find the integer solutions to $x^2-y^3=1$?

I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...

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### Solve in positive integers $n!=m(m+1)$

Is anybody know a solution of this problem? (Sorry, I've missed one summand in the previous post.)

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### Solve in positive integers $n!=m^2$

Is anybody know a solution of this problem?
(Sorry, correct question is here.)

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### An asymptotic expression for the solution to the squares problem suggested by statistical mechanics

The $s$ squares problem is to count the number $r_s (n)$ of integer solutions $(x_1,x_2,...,x_s)$ of the Diophantine equation $x_{1}^{2}+x_{2}^{2}+...+x_{s}^{2}=n$ in which changing the sign or ...

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### solution of the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$

i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.

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### Rational solutions of homogeneous equations

Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an ...

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### Fermat's Bachet-Mordell Equation

Fermat once claimed that the only integral solutions to $y^2 = x^3 - 2$ are $(3, \pm 5)$.
Fermat knew Bachet's duplication formulas (more precisely, Bachet had a formula for computing what we call ...

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### Trost's Discriminant Trick

The following little trick was introduced by E. Trost
(Eine Bemerkung zur Diophantischen Analysis,
Elem. Math. 26 (1971), 60-61). For showing that a diophantine equation
such as $x^4 - 2y^2 = 1$ ...

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### Is there any solution to the diophantine equation $1/x^m+1/y^n=1/z^t$ in which x, y, z are not coprime

If $x,y,z$ are coprime, this equation has not any solution with an elementary method. I want to know a solution of this equation when $x,y,z$ are not coprime.

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### Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...

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### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

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### Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1?

Let $S$ = { $a^2b^3$ : $a, b \in \mathbb{Z}_{>1}$ }.
Does there exist $n$ such that $n$, $n+1 \in S$?
Motivation: I was thinking about Question on consecutive integers with similar prime ...

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### Hexagonal Triangular Squares

Is there a hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously
(1) a perfect square, $n^2$, $n \ge 2$,
(2) a triangular number, ...

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### Integer polynomials taking square values

Is there a way to determine a formula giving all integer values of $x$ for which the value of a polynomial $P(x)$ with integer coefficients is a square?
That is, is there a closed formula for:
$X = ...

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### Checking local solubility of varieties at “bad” primes

Let $X$ be smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$ point, which can be lifted ...

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### Integer values of a rational function

Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the ...

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### Proving non-existence of solutions to $3^n-2^m=t$ without using congruences

I made a passing comment under Max Alekseyev's cute answer to this question and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was ...

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### $3^n - 2^m = \pm 41$ is not possible. How to prove it?

$3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?

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### Is there a solution for the equation x^m-y^n=k in which k > 1?

The Catalan conjecture state that $x^m-y^n=1$ has only the solution $x=3, m=2, y=2, n=3$. This conjecture was proved by Preda Mihailescu in 2004, but I want to know about the equation mentioned above. ...

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### Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?

The short version of my question is:
1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
2) For which positive ...

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### Status of Beal, Granville, Tijdeman-Zagier Conjecture [closed]

The Beal, Granville, Tijdeman-Zagier Conjecture, i.e.
If $A^x+B^y=C^z$ , where $A, B, C, x, y,z$ are positive integers and $x,
y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a ...

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### Diophantine equation: Egyptian fraction representations of 1

According to the OEIS (A002966) there are 294314 solutions in positive integers to the equation
$$\sum_{i=1}^7\frac{1}{x_i}=1$$ assuming $x_1\leq x_2\leq\cdots\leq x_7$.
Similarly for 8 summands there ...

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### Non-negative integer solutions of a single Linear Diophantine Equation

Consider the following linear Diophantine Equation::
ax + by + cz = d ------------ (1)
for all, a,b,c and d positive integers, and relatively prime, and ...

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### Quadratic Diophantine equations solver

Is there software that helps list small solutions of the Diophantine equation
$$
x_0^2=1+x_1^2+x_2^2+\cdots+ x_n^2
$$
where "small" is negotiable, but e.g. we could fix $x_0$ and and ask for the list ...

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### A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from ApĂ©ry's biography
(which some of the people have already enjoyed in
this answer).
During a mathematician's dinner in
Kingston, Canada, in 1979, the
conversation turned ...

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### Polynomials with rational coefficients

Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer ...